Assumptions of (a) Set Theory Carl Pollard Ohio State University Linguistics 680 Formal Foundations Thursday, September 23, 2010 These slides are available at: http://www.ling.osu.edu/ ∼ scott/680 1
Sets and Membership (1) • We assume there exist things called sets . • We assume there is a relationship, called membership , which, for any sets A and B , either does or does not hold between them. • If it does, we say A is a member of B , written A ∈ B . • If it doesn’t, we say A is not a member of B , written A / ∈ B . • We never say what sets are: they are the unanalyzed primitives of set theory and cannot be reduced to, or explained in terms of, more basic things that are not sets. 2
Assumptions about the Membership Relation (2) • We make certain further assumptions about membership. • Our set theory consists of these additional assumptions plus any conclusions we can draw from them using valid reasoning. • For now, we’ll state these assumptions informally in English. • Later we’ll state them more precisely in a special symbolic lan- guage (the language of set theory), and the precise restatements of the assumptions will be called the axioms of our set theory. • Also for now we don’t say exactly what counts as valid reasoning. • Later, we’ll specify what counts as valid reasoning in terms of mathematical objects called formal proofs which deduce new sen- tences (in the language of set theory) from the axioms. • There is nothing privileged about our set theory; there are other set theories which start with different assumptions. 3
Assumption 1 (Extensionality) (3) If A and B have the same members, then they are the same set (written A = B ). • We don’t mention that A and B are sets, because we’re doing set theory (so the only things we are talking about are sets). • We needn’t assume that if A and B do not have the same mem- bers, then they are not the same set (written A � = B ). That’s because, if they were the same set, then everything about them, including what members they had, would be the same. • If every member of A is a member of B , we say that A is a subset of B , written A ⊆ B . • If A ⊆ B , B might have members that are not in A . In that case we say A is a proper subset of B , written A � B . • But if both A ⊆ B and B ⊆ A , then it follows from Extension- ality that A = B . 4
Assumption 2 (Empty Set) (4) There is a set with no members. • From this assumption together with Extensionality we can con- clude that the there is only one set with no members. • We call this set the empty set, written ‘ ∅ ’. • But later, we’ll sometimes write it as ‘0’. • That’s because in the usual way of doing arithmetic within set theory (covered in Chapter 4) zero is the empty set. • As yet, we have no basis for concluding that there are any sets other than the empty set. • For example, we are not even able to make a valid argument that there is a set with ∅ as its only member. • We remedy this situation by making a few more assumptions. 5
Assumption 3 (Pairing) (5) For any sets A and B , there is a set whose members are A and B . • Even though we say ‘sets’ here, we don’t mean to rule out the possibility that A and B are the same set. • Because of Extensionality again, there is only one set whose members are A and B , which we write as { A, B } , or { B, A } . • More generally, we will notate any nonempty finite set by listing its members, separated by commas, between curly brackets, in any order. • We’ll get clear about what we mean by ‘finite’ in Chapter 5, but for now we’ll just rely on intuition. • In listing the members of a set, repititions don’t count, so e.g. if A and B are the same set, then { A, B } is the same set as { A } . • So it makes no sense to talk about how many times A is a mem- ber of B : either it is or it isn’t. 6
Definition (Singleton) (6) A singleton is a set with only one member. • For any set A ,we have enough resources now to prove informally (i.e. make a valid argument in English) that there is a singleton setwhose member is A . (Of course this is written ‘ { A } ’. • One singleton set is the set { 0 } whose member is 0. • { 0 } is also written ‘1’, because in the usual way of doing arith- metic within set theory, it is the same as the number one. 7
Preview of the Natural Numbers (7) • As mentioned above, we’ll define 0 to be ∅ , and 1 to be { 0 } . • By Pairing, we know there is a set, { 0 , 1 } , whose only members are 0 and 1. Let’s say that this is what the number 2 is. • There seems to be a pattern here, in which the next step would be to say that 3 is the set whose only members are 0, 1, and 2. • But as yet we don’t have sufficient resources to prove that there is such a set! • To say nothing of proving that there is a set whose members are the natural numbers. • In fact, as yet we don’t even know what ‘natural number’ means. • But soon we will (Chapter 4). 8
Assumption 4 (Union) (8) For any set A , there is a set whose members are those sets which are members of (at least) one of the members of A . • Extensionality ensures the uniqueness of such a set, which is called the union of A , written � A . If A = { B, C } , then � A is the set each of whose members is in • either B or C (or both), usually written B ∪ C . • In general, B ∪ C is not the same thing as { B, C } ! 9
Definition (Successor) (9) For any set A , the successor of A , written s ( A ), is the set A ∪ { A } . • That is, s ( A ) is the set with the same members as A , except that A itself is also a member of s ( A ). • Nothing in our set theory will rule out the possibility that A ∈ A , in which case s ( A ) = s ( A ). • However, some widely used set theories include an assumption (called Foundation ) which does rule out this possibility. • For example, we can prove that 1 is the successor of 0, and that 2 is the successor of 1. • Now we can say what 3 is: the successor of 2! • Likewise we can say what 4, 5, etc. are. • But we still can’t say exactly what we mean by a natural number. 10
Assumption 5 (Powerset) (10) For any set A , there is a set whose members are the subsets of A . • Again, Extensionality guarantees the uniqueness of such a set, which we call the powerset of A , written ℘ ( A ). • In general, ℘ ( A ) is not the same set as A , because usually the subsets of a set are not the same as the members of the set. • For example, 0 is a subset of 0 (in fact, every set is a subset of itself), but obviously 0 is not a member of 0 (since 0 is the empty set). 11
Assumptions vs. Definitions (11) a. Notice a crucial difference between the successor of a set A and the powerset of A : successor is defined in terms of things whose existence can already be established on the basis of previous assumptions (singletons, unions), whereas the existence of the powerset of A is assumed . b. Why didn’t we just define ℘ ( A ) to be the set whose members are the subsets of A ? c. It’s because nobody has found a valid argument (based on just the first four assumptions) that there is such a set! d. More generally, there is no guarantee that, for an arbitrary con- dition on sets P [ x ], there is a set whose members are all the sets x such that P [ x ]. e. But this fact did not become known till 1902. 12
Russell’s Paradox (12) a. Let P [ x ] be the condition ‘ x is not a member of itself’. b. Following Russell, we will show that there cannot be a set whose members are all the sets x such that P [ x ]. c. Suppose R were such a set. d. Then either (i) R is a member of itself, or (ii) it isn’t. i. Suppose R is a member of itself, Then it cannot be a member of R , since the members of R are sets which are not members of themselves. But then it is not a member of itself. ii. Suppose R is not a member of itself. Then it must be in R . But then, it is a member of itself. iii. Either way, assuming (c) leads to a contradiction. e. So the assumption (c) must have been false. 13
An Imaginable Set-Theoretic Assumption Bites the Dust (13) • Russell’s Paradox shows we don’t have the option of adding the following to our set theory: Tentative Assumption (Comprehension) For any condition P [ x ] there is a set whose members are all the sets x such that P [ x ]. • The following assumption is usually adopted instead. 14
Assumption 6 (Separation) (14) For any set A and any condition P [ x ], there is a set whose members are all the x in A that satisfy P [ x ]. • So far, assuming Separation has not been shown to lead to a contradiction. • Separation is so-called because, intuitively, we are separating out from A some members that are special in some way, and collecting them together into a set. • By Extensionality, there can be only one set whose members are all the sets x in A that satisfy P [ x ]. • We call that set { x ∈ A | P [ x ] } . 15
Intersection (15) • In naive introductions to set theory, the intersection of two sets A and B , written A ∩ B , is often ‘defined’ as the set whose members are those sets which are members of both A and B . • But how do we know there is such a set? • If we assume Separation and take P [ x ] to be the condition x ∈ B , then we can (unproblematically) define A ∩ B to be { x ∈ A | x ∈ B } . • A and B are said to intersect provided A ∩ B is nonempty. • A set A is called pairwise disjoint if no two distinct members of it intersect. 16
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